Problem 59 - Calculus Volume 1

Question

A unit circle is made up of n wedges equivalent to the inner wedge in the figure. The base of the inner triangle is 1 unit and its height is \(\sin \left(\frac{\pi}{n}\right) .\) The base of the outer triangle is \( B=\cos \left(\frac{\pi}{n}\right)+\sin \left(\frac{\pi}{n}\right) \tan \left(\frac{\pi}{n}\right) \) and the height is \(H=B \sin \left(\frac{2 \pi}{n}\right)\). Use this information to argue that the area of a unit circle is equal to \(\pi .\)

Step-by-Step Solution

Answer

The area of a unit circle is \( \pi \), proven as \( n \to \infty \).

Step 1: Area of Inner Triangle

The area of the inner triangle in the unit circle wedge can be calculated using the formula for the area of a triangle: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). For this inner triangle, the base is 1 unit and the height is \( \sin\left(\frac{\pi}{n}\right) \). Thus, the area is \( \frac{1}{2} \times 1 \times \sin\left(\frac{\pi}{n}\right) = \frac{1}{2} \sin\left(\frac{\pi}{n}\right) \).

Step 2: Area of Outer Triangle

For the outer triangle, we use the formula \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). The base of the outer triangle is \( B = \cos\left(\frac{\pi}{n}\right) + \sin\left(\frac{\pi}{n}\right) \tan\left(\frac{\pi}{n}\right) \) and the height is \( H = B \sin\left(\frac{2\pi}{n}\right) \). Thus, the area of the outer triangle is \( \frac{1}{2} \times B \times H = \frac{1}{2} B^2 \sin\left(\frac{2\pi}{n}\right) \).

Step 3: Sum of Areas for All Wedges

Since the unit circle is divided into \( n \) such wedges, the total area for all inner triangles in a full circle is \( \frac{n}{2} \sin\left(\frac{\pi}{n}\right) \). Similarly, the total area for all outer triangles in a full circle is \( \frac{n}{2} B^2 \sin\left(\frac{2\pi}{n}\right) \).

Step 4: Limit as n Approaches Infinity

As \( n \to \infty \), the length of each wedge becomes infinitesimally small. Therefore, the sum of the inner areas approaches the area of a circle, which is \( \pi \). The extra area from the outer triangles diminishes to zero. This results in the total area being \( \pi \).

Step 5: Conclusion

Through this limit process, it is reasonable to conclude that the sum of the areas of these wedges, both inner and outer, tends to the area of the circle as \( n \to \infty \). Thus, the calculated total area of the unit circle is \( \pi \).

Key Concepts

Area of a Circle

Understanding the area of a circle is essential in geometry and calculus. The area of a unit circle—where the radius is 1—is simply Pi (\(\pi\)), which represents the total space within the circle's boundary.
To visualize this, imagine breaking the circle into multiple tiny wedges or segments. By calculating the area of each wedge and summing them up, you can approximate the circle's area. When you increase the number of wedges, approaching infinity, these sums converge precisely to the actual area of the circle, which is \(\pi\).
This method of summing the area of wedges is not only a fascinating approach to finding the circle's area but also demonstrates the beauty and interconnectedness of geometry and calculus.

Trigonometric Identities

Trigonometry provides us with valuable tools, especially when dealing with circles and angles. In the context of a unit circle, trigonometric identities simplify the calculation process, especially when determining lengths and heights of triangles that make up a circle.
Key identities include:

• The sine function, \(\sin(\theta)\), which gives us the height of a triangle in the unit circle context.
• The cosine function, \(\cos(\theta)\), which helps determine the base length.
• The tangent function, \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), assists in calculating other significant dimensions like the base of the outer triangle.

By leveraging these identities, we can efficiently divide circles into manageable parts, simplifying complex geometry problems.

Limit Process

The limit process is a fundamental concept in calculus that allows us to understand how functions behave as inputs approach certain values. In our example of finding the circle's area, the limit process is used to determine what happens as the number of wedges, \(n\), increases indefinitely.
As \(n\) tends toward infinity:

• Each wedge becomes narrower and the sum of their areas approaches the area of the circle.
• The additional area from the outer triangles grows negligible, meaning we capture the exact circle's area without any excess.

This convergence illustrates a key property of circles: as the number of divisions increases, the approximation improves, ultimately equaling the exact area—an elegant application of limits in real-world problems.

Geometry

Geometry provides the necessary foundation for understanding shapes and spaces, such as circles. In the unit circle problem, geometric principles allow us to assess and validate the spatial reasoning behind dividing a circle into smaller triangles.
Some geometric concepts essential to solving the exercise include:

• Understanding triangles: Each wedge is considered a triangle, where one knows the base and needs to determine the height.
• Area calculation: By knowing the triangle's base and height, we apply the area formula \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\) to each section.
• Interplay of angles and arcs: Recognizing that the circle's total angle (360 degrees or \(2\pi\) radians) affects how wedges are formed and contribute to the overall structure of the circle.

The seamless integration of these geometric concepts allows us to dissect and analyze the circle comprehensively, making the calculations intuitive and precise.